3 Computing option prices under stochastic volatility models
3.4 Pricing options with the filtration consisting of prices
In this section, a new scheme for inference and pricing under the stochastic volatility models is proposed. The above two subtle difficulties may be overcome with this approach.
3.4.1 An algorithm for ML estimation of parameters and option pricing
In contrast to the stochastic volatility models, GARCH models provide as a tool that are feasible in practical operations. There are some facts about the two types of models to be pointed out.
1. For specific sequence of observations, the joint distributions of GARCH models are analytical, while generally the stochastic volatility models are not.
2. As the period of observations approaches 0, the GARCH models converge to stochastic volatility models in distribution, see Nelson (1990), Duan (1997) and Fornari and Mele (2004).
The first point implies that likelihood inference with the stochastic volatility models is generally infeasible. However, in light of the second argument, approximating
stochastic volatility models with some specially designed GARCH is possible.
For this point, Brown et al. (2003) show the asymptotic equivalence of GARCH and stochastic volatility models. And the partially observed GARCH with MCMC methods for statistical inference is proposed in chapter 2.
Let
( X
~(1),V
~(1))
be drawn from the stochastic volatility model and( Y
~(1),h
~(1))
fromthe corresponding GARCH with length of construction interval Δ. Denote
(
−+Δ ⋅ ⋅⋅ −Δ= t t
t Y Y
Y~ 1 , ,
)
* as the augmented data between Yt and Yt-1 . Note that the observed data is indeed ~(1)
X
, or equivalently ~(1)Y
. Past paths Y*~conditional on the observed data ~(1)
Y
can be generated with the following algorithm.1. Initialize V0 and *~(Δ)
Y
. 2. Updateh
from h |*Y~.3. Update sequentially*Y~t(Δ) from ~ ;η
4. Repeat step 2 and 3 until convergence and take L independent paths.
The likelihood function can be approximated as
∑ ∏
By maximizing the above function, maximum likelihood estimator for the parameters can be obtained. Option prices then can be calculated based on the estimated
parameters and the L independent paths.
5. Extending the L paths using general Monte Carlo methods with the risk-neutral measure under the GARCH model to get L samples of YT and thus the option prices.
This method converts the bivariate diffusion process with the second process
unobservable into a partially observed univariate GARCH process. So it does provide an approximation for the formulation of the option prices (3.3), without requiring the estimation of the current variance, Vt or ht. It is also applicable to various types of derivatives as long as the payoff function at maturity involves the prices only.
Furthermore an important implication is that the option prices at time t may depend on (S0, ···, St) in a much more complicated manner, instead of the estimated volatilities only. This important characteristic provides a way to distinct valuations of options conditional on different patterns of past paths. Next, a simple numerical experiment will be conducted for further studies.
3.4.2 Empirical performance of the algorithm
First, 12 disjoint paths whose variances at terminal ranges from 0.004 to 0.026 are selected. Both the Nelson and Foster filter and the partially observed GARCH methods are applied to estimate the variances. For the partially observed GARCH method, each observation period of 0.004 is divided into 5 subdivisions.
Metropolis-Hasting algorithm is applied to each segment with 200 iterations of
burn-in. After 100 iterations for the whole path, 1200 samples of paths are taken every two iterations.
Since there exists a monotone relation between the options prices, it suffices to compare the estimated variances at time t. From Figure 3.3, it can be seen that the
Nelson and Foster filter actually provides as an asymptotic unbiased estimator, although it tends to have larger variance. In contrast, the partially observed GARCH estimates seem to be less volatile but biased upward at lower and intermediate level of the true variance at terminal point.
Figure 3.3: Comparison between the filters.
0.000 0.005 0.010 0.015 0.020 0.025 0.030
0.000 0.005 0.010 0.015 0.020 0.025 0.030
Nelson filter partially observed GARCH - mean true variance
Figure 3.4: Conditional distribution of Vt given (S0,···, St)′ for the path with terminal point having level of variance about 0.012.
0 40 80 120 160
0 0.005 0.01 0.015 0.02 0.025 0.03
ht
p(ht|X)
Partial Observed GARCH Nelson filter
The observed unbiasedness means that the option prices obtained with the Nelson and Foster filter essentially match the prices by assuming that the true variances are known. On the other hand, the option prices from the partially observed GARCH tend to be higher compared to the “true” prices as the level of the true variance at terminal is under or near the long term equilibrium level of the variance process. The source of the bias may be partly due to the discretization error. But more probable, it may indicate that the conditional distribution of V given (S ,···, S)’ is positively biased as
in Figure 3.3. The property coincides with the phenomenon that the implied volatilities are generally higher than the historical volatilities or most of
model-implied volatilities. Of course, more details are worth further investigations.
Next, from the very long simulated path, 10 ascent and 10 descent disjoint paths of length 100 are selected according to the conditions below:
1. The variance at the last point is between 0.0097 and 0.013.
2. The absolute change rate through the 100 periods is the largest.
Figure 3.5: Ascent and descent paths.
Ten ascent paths
4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65
1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97
Ten descent paths
4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80
1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97
These paths are shown in Figure 3.5. And in Figure 3.6, the true volatilities, the
Nelson and Foster filter and partially GARCH, implied volatilities are plotted together.
Again, the unbiasedness of the Nelson and Foster filter is revealed, indifferent to ascent or descent paths. However, in addition to the positive bias, the partially observed GARCH implied volatilities are definitely dependent on the past paths.
There clearly exist significant differences between the volatilities from ascent paths and descent paths. This property certainly may help to explain the asymmetry of the markets shown in Figure 3.2.
Figure 3.6: Variance estimates for ascent and descent paths.
0.000 0.005 0.010 0.015 0.020 0.025
u01 u02 u03 u04 u05 u06 u07 u08 u09 u10 d01 d02 d03 d04 d05 d06 d07 d08 d09 d10
Nelson filter partially observed GARCH - mean true variance
In addition to the consistency in theory and integration in practical operation, the results summarized above show that the partially observed GARCH approach provides huge potential to rationalize the behavior of the participants in the option markets, while the traditional method does not. A major key is of course the fact that volatilities are unobservable and should not simply be estimated. Furthermore, the method leads to a one-to-one relation between the risk premium of volatilities and option prices. As recently many researches are focused on the dynamics of implied volatility surfaces, alternative choice such as premium surface can be explored to further understanding of the behavior of the financial markets.